Assuming N terminations of channels form a uniform linear array (ULA), x =[x₁, x₂...x_N]^T is the signal input, t is time, w =[w₁, w₂...w_N]^T are corresponding weights, and the signal model is expressed as

(1)

Then the output composed by the ULA can be given by

(2)

Where (.)^H denotes the conjugate transpose, (.)^T denotes transpose, n(t) is the random noise, s₀ (t) and i(t) are the desired signal and unwanted interference, respectively. So the output power can be obtained as

(3)

From (2) and (3), we can get the mean output power of the array system expressed as

(4)

Where E[.] denotes the expectation operator. The correlation matrix R is defined as

(5)

Generally, it is assumed that the array system platform is located in the far field of targets, and then the directional signal from scattering of targets is induced with array as plane wave. By the array antenna theory, the k-th source measured at one of the element of array is given by

(6)

As definition in [12L. Ehrenberg, S. Gannot, A. Leshem, and E. Zehavi, "Sensitivity analysis of MVDR and MPDR beamformers", IEEE 26^th Convention of Electrical and Electronics Engineering, Israel, 2010.
[http://dx.doi.org/10.1109/EEEI.2010.5662190] ], the steering vector of echoes are represented as

(7)

Where is the phase delay. By introducing (4) and (7) into (5), correlation matrix R can be simply rewritten as

(8)

Where S_o is the desired signal steering vector, and are the power of the uncorrelated impinging desired signal, interference and noise, respectively.

From the aforementioned analysis, the array system output power can be expressed as

(9)

1) Beam Pattern

Weight of CBF is a static response to the elements of the array system. In contrast, the weight of MVDR is adaptive on the array sample data, and it can be optimized with maximum impulse response to the desired direction of arrival. In this beamformer, array system can achieve minimum output power and maximum signal-interference-noise ratio (SINR). Generally, CBF's weight is defined as

(10)

MVDR’s weight can be derived from the following linearly constrained quadratic problem [13D.L. Duttweiler, "Proportionate normalized least-mean-squares adaptation in echo cancelers", IEEE Trans. Speech Audio Process., vol. 8, no. 5, pp. 508-518, 2000.
[http://dx.doi.org/10.1109/89.861368] ] as

(11)

Then the solution of (11) by least square method is performed as

(12)

By referring to [14L.C. Godara, Smart Antenna., CRC Press: Boca Raton, FL, 2004, pp. 29-30.
[http://dx.doi.org/10.1201/9780203496770] ] and using matrix Inversion Lemma, the correlation matrix R can be replaced by data snapshots and expressed as

(13)

Where n is snapshot number of data, N is channel number.

2) Output Power and Array Gain

Assuming that noise and interference are uncorrelated, by introducing (10) and (12) into (9), respectively, the output power for both CBF and MVDR beamformer is expressed as (14) and (15), respectively

(14)

(15)

Similarly, SINR for both CBF and MVDR beamformer can be obtained as [15J. Ward, H. Cox, and S. Kogon, "A comparison of robust adaptive beamforming algorithms", In: Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, California, USA, 2003, pp. 1340-1344.
[http://dx.doi.org/10.1109/ACSSC.2003.1292206] ]

(16)

(17)

According to the [16H.L. Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory., Wiley: New York, 2002, pp. 1200-1210.
[http://dx.doi.org/10.1002/0471221104] ], array gain for both CBF and MVDR beamformer can be derived as (18) and (19), respectively

(18)

(19)

Fig. (1) shows the beam pattern of output signal about CBF and MVDR beamformer. The case is that the degree of arrival (DOA) of clutter is ±40º, and the steering direction is 0º, number of linear array antenna is 8, central frequency of half wavelength antenna is 150 MHz, and element distance is half wavelength. As shown in Fig. (1), MVDR method has a deeper null level of interference degree and performs better than CBF more than 30 dB. It means that MVDR beamformer can decrease the clutter at the interference degree.

Fig. (1) Beam pattern of CBF and MVDR beamformer.

Assuming that the error happens to weights of beamformer and follows statistics as (20), and mean output power and SINR can be analyzed by [10L.C. Godara, "Error analysis of the optimal antenna array processors", IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, no. 4, pp. 395-409, 1986.
[http://dx.doi.org/10.1109/TAES.1986.310775] ]

(20)

(21)

(22)

By introducing (10), (12), (21) and (22) into (9), the output power of signal can be rewritten as (23) and (24), respectively

(23)

(24)

Taking the mean value on both sides in (23) and (24), the mean output power of CBF and MVDR beamformer about signal can be expressed as (25) and (26), respectively

(25)

(26)

Similarly, the mean output power of CBF and MVDR beamformer about interference and noise are given by (27) and (28), respectively

(27)

(28)

Obviously, the output SINR of CBF and MVDR beamformer can be defined as (29) and (30)

(29)

(30)

To study the error effects of beamformer, the SINR has been compared by both CBF and MVDR method with the same parameters of array system. To simply analytical process, assuming, the input SIR is equal to 0 dB, 20 dB and 40 dB, respectively. Weight fluctuation is considered to follow statistics characteristics as (20). In Figs. (2 and 3), the output SINR of CBF and MVDR beamformer with different input SIR has been simulated. It shows that output SINR of array system becomes worse with weight fluctuating increasing, and they have maximum SINR when errors of weight are equal to zero. Compared to CBF beamformer, the SINR of MVDR beamformer has a higher value when weight error is equal to zero. They have the same tendency of SINR as weight error changes. The output SINR is down from 30 dB to 0 dB with weight error increasing from 0 to 0.2. It is obvious that the SINR of both MVDR and CBF beamformer have better performance when the input SIR gets bigger. It means that the array system could get higher output SINR with the input SIR increasing. However, they both perform not well when variance of weight becomes larger. Kinds of reasons may cause the weight vector error, such as uncertainty in the positions of array elements, assuming the plane wave arrives at the array and makes errors in the steering vector or phase. The numbers of snapshots for calculating the correlation matrix may also cause weight errors. The output SINR is sensitive to the weight fluctuation from the simulation in Fig. (2) and Fig. (3). It indicates that both MVDR and CBF beamformer should get accurate weight to avoid output SINR becoming worse.

Fig. (2) Output SINR of CBF beamformer with error of weight vector.

Fig. (3) Output SINR of MVDR beamformer with error of weight vector.

One reason for errors of steering vector is that the array system of platform can’t be steady all the time especially in high-altitude radar for air turbulence. Assuming that variance of steering vector is distributed the same as (20), both CBF and MVDR beamformer have the same steering vector errors with correlation spectral . From (9), we can get the desired signal and clutter mean output power of CBF beamformer expressed as

(31)

(32)

Where Then SINR and gain of CBF beamformer can be given by

(33)

(34)

Signal-Plus-Noise matrix inverse processor is used for calculating weight of MVDR and it can be also replaced by Noise matrix inverse as previously mentioned. Defining, denotes the trace of . The output SINR and gain of MVDR beamformer can be expressed as

(35)

(36)

The output SINR for both CBF and MVDR beamformer with error of steering vector is shown in Figs. (4 and 5). It can be seen that the output SINR of CBF beamformer declines slowly as variance of steering vector becomes larger. In contrast, output SINR of MVDR beamformer declines quickly with linear change as steering vector variance changes from 0 to 0.2. It means that the MVDR method is more sensitive to variance of steering vector than that of CBF beamformer. Moreover, From (33) and (35) one can know that the denominator of SINR_MVDR has which does not exist in SINR_CBF. The algebraic term leads to the output SINR of MVDR drops quickly.

Fig. (4) Output SINR of CBF beamformer with error of steering vector.

Fig. (5) Output SINR of MVDR beamformer with error of steering vector.